Problem: $f(x) = \dfrac{ \sqrt{ x - 8 } }{ x^2 + 15 x + 54 }$ What is the domain of the real-valued function $f(x)$ ?
Solution: $f(x) = \dfrac{ \sqrt{ x - 8 } }{ x^2 + 15 x + 54 } = \dfrac{ \sqrt{ x - 8 } }{ ( x + 9 )( x + 6 ) }$ First, we need to consider that $f(x)$ is undefined anywhere where the radical is undefined, so the radicand (the expression under the radical) cannot be less than zero. So $x - 8 \geq 0$ , which means $x \geq 8$ Next, we also need to consider that $f(x)$ is undefined anywhere where the denominator is zero. So $x \neq -9$ and $x \neq -6$ However, these last two restrictions are irrelevant since $8 > -9$ and $8 > -6$ and so $x \geq 8$ will ensure that $x \neq -9$ and $x \neq -6$ Combining these restrictions, then, leaves us with simply $x \geq 8$ Expressing this mathematically, the domain is $\{ \, x \in \RR \mid x \geq8\, \}$.